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A317532
Regular triangle read by rows: T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.
12
1, 2, 2, 4, 8, 4, 8, 34, 26, 8, 16, 124, 168, 76, 16, 32, 448, 962, 674, 208, 32, 64, 1568, 5224, 5344, 2392, 544, 64, 128, 5448, 27336, 39834, 24578, 7816, 1376, 128, 256, 18768, 139712, 283864, 236192, 99832, 24048, 3392, 256, 512, 64448, 702496, 1960320, 2161602, 1186866, 370976, 70656, 8192, 512
OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
EXAMPLE
The T(3,2) = 8 multiset partitions:
{{1},{1,1}}
{{1},{2,2}}
{{2},{1,2}}
{{1},{1,2}}
{{2},{1,1}}
{{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
Triangle begins:
1
2 2
4 8 4
8 34 26 8
16 124 168 76 16
32 448 962 674 208 32
...
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n], Length[#]==k&]], {n, 7}, {k, n}]
PROG
(PARI) \\ here B(n, k) is A239473(n, k).
B(n, k)={sum(r=k, n, binomial(r, k)*(-1)^(r-k))}
Row(n)={Vecrev(sum(j=1, n, B(n, j)*polcoef(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^binomial(k+j-1, j-1)), n))/y)}
{ for(n=1, 10, print(Row(n))) } \\ Andrew Howroyd, Dec 31 2019
CROSSREFS
Row sums are A255906.
Sequence in context: A213418 A317517 A300182 * A222659 A116694 A220810
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jul 30 2018
EXTENSIONS
Terms a(29) and beyond from Andrew Howroyd, Dec 31 2019
STATUS
approved